Draw a neat and labelled energy level diagram and explain Balmer series and Brackett series of spectral lines for hydrogen atom.

In figure, horizontal lines represent energy levels for different orbits. The vertical arrow lines show various electronic transitions. There are different series of spectral lines for hydrogen atom. Some of them are Balmer series and Brackett series. Balmer Series: This series originates due to transitions of the electrons from different outer orbits to the second orbit ( p = 2 ) . Therefore, the wavelength of Balmer series is given by : 1 λ = R 1 2 2 - 1 n 2 where, n = 3 , 4 , 5 . . . . This series lies in the visible region of the spectrum. Brackett Series: This series arises due to transitions of the electrons from different outer orbits to the fourth orbit ( p = 4 ) . The wavelength of Brackett series is given by : 1 λ = R 1 4 2 - 1 n 2 ; where, n = 5 , 6 , 7 . . . . This series lies in near infrared region of the spectrum.

Calculate the de-Broglie wavelength of an electron moving with one fifth of the speed of light. Neglect relativistic effects.( h = 6 . 63 × 10 - 34 J s , c = 3 × 10 8 m s - 1 , mass of electron, m e = 9 × 10 - 31 kg )

By de-Broglie's theory, the wavelength of a particle is given by λ = h P = h m v . Given : Planck constant, h = 6 . 63 × 10 - 34 J s . Speed of light, c = 3 × 10 8 m s - 1 . Mass of electron, m e = 9 × 10 - 31 kg . It travels in one fifth of the speed of the speed of the light. Wavelength of the electron, λ = h m v = 6 . 63 × 10 - 34 9 × 10 - 31 × c 5 = 6 . 63 × 10 - 34 9 × 10 - 31 × 3 × 10 8 5 = 1 . 227 × 10 - 11 m .

State the law of radioactive decay. Hence derive the relation N = N 0 e - λt . Represent it graphically.

Law of radioactive decay : At any instant, the rate of radioactive disintegration is directly proportional to the number of nuclei of the radioactive element present at that instant. We know by radioactive decay law: d N d t ∝ N ⇒ d N d t = - λ t where λ is the constant of proportionality called radioactive decay constant. Let N 0 be the number of nuclei present at time t = 0 , and N the number of nuclei present at time t . ∫ N 0 N d N N = ∫ 0 t - λ d t = - λ ∫ 0 t d t ⇒ ln N - ln N 0 = - λ t ⇒ ln N N 0 = - λ t ⇒ N = N 0 e - λ t Graphically, it can be represented as: